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HEX OF MATHEMATICS
metric functions had first presented' themselves this way.
Would it not have been 'more natural' to consider the inverses
of these functions, that is, the familiar trigonometric functions
themselves as the given functions to be studied and analysed?
Undoubtedly; but in shoals of more advanced problems, the
simplest of which is that of finding the length of the arc of an
ellipse by the integral calculus, the awkward inverse 'elliptic'
(not 'circular', as for the arc of a circle) functions presented
themselves first It took Abel to see that these functions should
be 'inverted* and studied, precisely as in the case of sin x, cos aj
instead of sin"1 x9 cos"1 r. Simple, was it not? Yet Legendre, a
great mathematician, spent more than forty years over his
"elliptic integrals' (the awkward 'inverse functions* of his
problem) without ever once suspecting that he should invert.*
This extremely simple, uncommonsensical way of looking at an
apparently simple but profoundly recondite problem was one
of the greatest mathematical advances of the nineteenth
century.
All this however was but the beginning, although a suffi-
cientlv tremendous beginning  like Kipling's dawn coming up
like thunder - of what Abel did hi his magnificent theorem and
in his work on elliptic functions. The trigonometric or circular
functions have & single real period, thus sin (as + 2ir) = sin #,
etc. Abel discovered that his new functions provided by the
inversion of an elliptic integral have precisely two periods,
whose ratio is imaginary. After that, Abel's followers in this
direction - Jacobi, Rosenhain, Weierstrass, Riemann, and
many more - mined deeply into Abel's great theorem and by
carrying on and extending his ideas discovered functions of n
variables having 2n periods. Abel himself carried the exploita-
tion of his discoveries far. His successors have applied all this
work to geometry, mechanics, parts of mathematical physics,
and other tracts of mathematics, solving important problems
* In ascribing priority to Abel, rather than 'joint discovery' to Abel
and Jaoobi, in this matter, I have followed Mittag-Leffler. From a
thorough acquaintance with all the published evidence, J am con-
vinced that AbeFs claim is indisputable, although Jacobi's com-
patriots argue otherwise.
35G